MathDB

2013 Stanford Mathematics Tournament

Part of Stanford Mathematics Tournament

Subcontests

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2013 General Problem 15

Given regular hexagon ABCDEFABCDEF, compute the probability that a randomly chosen point inside the hexagon is inside triangle PQRPQR, where PP is the midpoint of ABAB, QQ is the midpoint of CDCD, and RR is the midpoint of EFEF.

SMT 2013 Team #15

Suppose we climb a mountain that is a cone with radius 100100 and height 44. We start at the bottom of the mountain (on the perimeter of the base of the cone), and our destination is the opposite side of the mountain, halfway up (height z=2z = 2). Our climbing speed starts at v0=2v_0=2 but gets slower at a rate inversely proportional to the distance to the mountain top (so at height zz the speed vv is (h−z)v0/h(h-z)v_0/h). Find the minimum time needed to get to the destination.
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2013 General Problem 13

R2\mathbb{R}^2-tic-tac-toe is a game where two players take turns putting red and blue points anywhere on the xyxy plane. The red player moves first. The first player to get 33 of their points in a line without any of their opponent's points in between wins. What is the least number of moves in which Red can guarantee a win? (We count each time that Red places a point as a move, including when Red places its winning point.)

SMT 2013 Team #13

A board has 22, 44, and 66 written on it. A person repeatedly selects (not necessarily distinct) values for xx, yy, and zz from the board, and writes down xyz+xy+yz+zx+x+y+zxyz+xy+yz+zx+x+y+z if and only if that number is not yet on the board and is also less than or equal to 20132013. This person repeats this process until no more numbers can be written. How many numbers will be written at the end of the process?
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